Definition:Confidence Interval/Definition 2
Definition
Let $X$ be a random variable.
Let $\theta$ be a population parameter of $X$ whose distribution is unknown.
A $100 \paren {1 - \alpha}$ percent confidence interval for $\theta$ is an interval formed by a rule which ensures that, in the long run, $100 \paren {1 - \alpha}$ percent of such intervals will include $\theta$.
This confidence interval is derived from the information obtained from a random sample of observations of $X$.
Confidence Limit
The endpoints of a confidence interval are known as confidence limits.
Examples
$95 \%$ Confidence Interval
A $95 \%$ confidence interval is a confidence interval whose $\alpha$ parameter is:
- $\alpha = 0 \cdotp 05$
Let $\bar x$ be the mean of a sample of $n$ observations from a normal distribution with unknown mean $\mu$ and known standard deviation $\sigma$.
Then a $95 \%$ confidence interval for $\mu$ is:
- $\closedint {\bar x - \dfrac {1 \cdotp 96 \sigma} {\sqrt n} } {\bar x + \dfrac {1 \cdotp 96 \sigma} {\sqrt n} }$
Motivation
A $100 \paren {1 - \alpha}$ percent confidence interval for a population parameter $\theta$ derived from a given sample covers all values of $\theta_0$ of that parameter that would be accepted at significance level $\alpha$ in a hypothesis test of:
- the null hypothesis $H_0: \theta = \theta_0$
against:
- the alternative hypothesis $H_1: \theta \ne \theta_0$.
Also see
- Results about confidence intervals can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): confidence interval
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): estimation
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): confidence interval
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): estimation